Abelian Divisibility Sequences Joseph H. Silverman Brown University Arithmetic of Low-Dimensional Abelian Varietes ICERM, June 3–7 2019 0
Classical Divisibility Sequences 1 Divisibility Sequences A divisibility sequence is a sequence of (positive) integers ( D n ) n ≥ 1 such that m | n = ⇒ D m | D n .
Classical Divisibility Sequences 1 Divisibility Sequences A divisibility sequence is a sequence of (positive) integers ( D n ) n ≥ 1 such that m | n = ⇒ D m | D n . Classical examples divisibility sequences include: D n = a n − b n , where a > b ≥ 1; the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, . . . .
Classical Divisibility Sequences 1 Divisibility Sequences A divisibility sequence is a sequence of (positive) integers ( D n ) n ≥ 1 such that m | n = ⇒ D m | D n . Classical examples divisibility sequences include: D n = a n − b n , where a > b ≥ 1; the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, . . . . An elliptic divisibility sequence ( EDS ) is formed from an elliptic curve E/ Q and a non-torsion point P ∈ E ( Q ) by writing � A n ( P ) � D n ( P ) 2 , B n ( P ) nP = . D n ( P ) 3 � � The sequence D n ( P ) n ≥ 1 is an EDS.
Classical Divisibility Sequences 2 Divisibility Sequences over Dedekind Domains More generally, if R is a Dedekind domain, we define an R -divisibility sequence to be a sequence of ideals ( D n ) n ≥ 1 such that m | n = ⇒ D m | D n .
Classical Divisibility Sequences 2 Divisibility Sequences over Dedekind Domains More generally, if R is a Dedekind domain, we define an R -divisibility sequence to be a sequence of ideals ( D n ) n ≥ 1 such that m | n = ⇒ D m | D n . In this way we can define an EDS, for example, by fac- toring the ideal generated by x ( nP ) in the form x ( nP ) R = A n ( P ) D n ( P ) − 2 and taking the sequence � � D n ( P ) n ≥ 1 .
Classical Divisibility Sequences 3 Reformulating EDS Let E/K be an elliptic curve, let P ∈ E ( K ), and let E /R be a N´ eron model for E/K . Then the EDS � � D n ( P ) n ≥ 1 is characterized by noting that for each prime ideal p of R , we have ∗ � � largest k so that nP ≡ O (mod p k ) ord p D n ( P ) = . ∗ Maybe not quite right at primes of bad reduction.
Classical Divisibility Sequences 3 Reformulating EDS Let E/K be an elliptic curve, let P ∈ E ( K ), and let E /R be a N´ eron model for E/K . Then the EDS � � D n ( P ) n ≥ 1 is characterized by noting that for each prime ideal p of R , we have ∗ � � largest k so that nP ≡ O (mod p k ) ord p D n ( P ) = . ∗ Maybe not quite right at primes of bad reduction. Or we can simply say that D n ( P ) is the largest ideal (ordered by divisibility) such that nP ≡ O (mod D n ( P )) .
Classical Divisibility Sequences 4 Abelian Divisibility Sequences: Type I In general: R a Dedekind domain. K the fraction field of R . A/K an abelian variety. A /R a N´ eron model for A/K . P ∈ A ( K ) a non-torsion point. The abelian divisibility sequence ( ADS ) for the � � pair ( A, P ) is the sequence of ideals D n ( P ) n ≥ 1 defined by the property that D n ( P ) is the largest ideal satisfying nP ≡ O (mod D n ( P )) .
Classical Divisibility Sequences 4 Abelian Divisibility Sequences: Type I In general: R a Dedekind domain. K the fraction field of R . A/K an abelian variety. A /R a N´ eron model for A/K . P ∈ A ( K ) a non-torsion point. The abelian divisibility sequence ( ADS ) for the � � pair ( A, P ) is the sequence of ideals D n ( P ) n ≥ 1 defined by the property that D n ( P ) is the largest ideal satisfying nP ≡ O (mod D n ( P )) . Alternatively, letting π : A → Spec( R ), we can de- fine D n ( P ) via arithmetic intersection theory, D n ( P ) := π ∗ ( nP · O ) = ( nP ) ∗ ( O ) ∈ Div � � Spec( R ) .
Classical Divisibility Sequences 4 Abelian Divisibility Sequences: Type I In general: R a Dedekind domain. K the fraction field of R . A/K an abelian variety. A /R a N´ eron model for A/K . P ∈ A ( K ) a non-torsion point. The abelian divisibility sequence ( ADS ) for the � � pair ( A, P ) is the sequence of ideals D n ( P ) n ≥ 1 defined by the property that D n ( P ) is the largest ideal satisfying nP ≡ O (mod D n ( P )) . Alternatively, letting π : A → Spec( R ), we can de- fine D n ( P ) via arithmetic intersection theory, D n ( P ) := π ∗ ( nP · O ) = ( nP ) ∗ ( O ) ∈ Div � � Spec( R ) . � � Exercise : Prove that D n ( P ) is a divisibility sequence.
Classical Divisibility Sequences 5 Growth Rates A G m -divisibility sequence D = ( D n ) such as a n − b n or the Fibonacci sequence grows exponentially, n →∞ | D n | 1 /n > 1 . lim
Classical Divisibility Sequences 5 Growth Rates A G m -divisibility sequence D = ( D n ) such as a n − b n or the Fibonacci sequence grows exponentially, n →∞ | D n | 1 /n > 1 . lim � � Elliptic divisiblity sequences D = D n ( P ) grow even faster, � 1 /n 2 � lim N K/ Q D n ( P ) > 1 . ( ∗ ) n →∞ Two remarks about elliptic divisibility sequences: • The limit in ( ∗ ) is ˆ H E ( P ), i.e., H E ( P ) n 2 = ˆ N K/ Q D n ( P ) ≈ ˆ H E ( nP ) . • The proof uses a deep, ineffective theorem of Siegel.
Classical Divisibility Sequences 5 Growth Rates A G m -divisibility sequence D = ( D n ) such as a n − b n or the Fibonacci sequence grows exponentially, n →∞ | D n | 1 /n > 1 . lim � � Elliptic divisiblity sequences D = D n ( P ) grow even faster, � 1 /n 2 � lim N K/ Q D n ( P ) > 1 . ( ∗ ) n →∞ Two remarks about elliptic divisibility sequences: • The limit in ( ∗ ) is ˆ H E ( P ), i.e., H E ( P ) n 2 = ˆ N K/ Q D n ( P ) ≈ ˆ H E ( nP ) . • The proof uses a deep, ineffective theorem of Siegel. The height of nP on an abelian variety grows at a sim- ilar rate, but co-dimension considerations suggest that D n ( P ) might not grow that fast.
Classical Divisibility Sequences 6 Growth Rates of ADS: A Conjecture Let A/K be an abelian variety of Conjecture 1. dimension ≥ 2, and let P ∈ A ( K ) be a point such that Z P is Zariski dense in A . Then � 1 /n 2 � lim N K/ Q D n ( P ) = 1 . n →∞
Classical Divisibility Sequences 6 Growth Rates of ADS: A Conjecture Let A/K be an abelian variety of Conjecture 1. dimension ≥ 2, and let P ∈ A ( K ) be a point such that Z P is Zariski dense in A . Then � 1 /n 2 � lim N K/ Q D n ( P ) = 1 . n →∞ The conjecture says that in dimension ≥ 2, an ADS grows more slowly than the heights of the points in the sequence nP .
Classical Divisibility Sequences 6 Growth Rates of ADS: A Conjecture Let A/K be an abelian variety of Conjecture 1. dimension ≥ 2, and let P ∈ A ( K ) be a point such that Z P is Zariski dense in A . Then � 1 /n 2 � lim N K/ Q D n ( P ) = 1 . n →∞ The conjecture says that in dimension ≥ 2, an ADS grows more slowly than the heights of the points in the sequence nP . Theorem. Conjecture 1 follows from Vojta’s conjec- ture applied to A blown up at O .
Classical Divisibility Sequences 7 A Multiplicative Analogue to Conjecture 1 Here is a G m analogue. We replace A by G 2 m and P ∈ A ( K ) with ( a, b ) ∈ G 2 m ( Q ). The associated divisibility sequence gcd( a n − 1 , b n − 1) measures the “arithmetic distance” from ( a, b ) n to (1 , 1).
Classical Divisibility Sequences 7 A Multiplicative Analogue to Conjecture 1 Here is a G m analogue. We replace A by G 2 m and P ∈ A ( K ) with ( a, b ) ∈ G 2 m ( Q ). The associated divisibility sequence gcd( a n − 1 , b n − 1) measures the “arithmetic distance” from ( a, b ) n to (1 , 1). (Bugeaud–Corvaja–Zannier 2003) Let Theorem. a, b ∈ Z with | a | > | b | > 1. Then n →∞ gcd( a n − 1 , b n − 1) 1 /n = 1 . lim ([BCZ] result is more general. See also work of A. Levin.)
Classical Divisibility Sequences 7 A Multiplicative Analogue to Conjecture 1 Here is a G m analogue. We replace A by G 2 m and P ∈ A ( K ) with ( a, b ) ∈ G 2 m ( Q ). The associated divisibility sequence gcd( a n − 1 , b n − 1) measures the “arithmetic distance” from ( a, b ) n to (1 , 1). (Bugeaud–Corvaja–Zannier 2003) Let Theorem. a, b ∈ Z with | a | > | b | > 1. Then n →∞ gcd( a n − 1 , b n − 1) 1 /n = 1 . lim ([BCZ] result is more general. See also work of A. Levin.) The proof uses Schmidt’s subspace theorem and is sur- prisingly intricate, even for a = 3 and b = 2. Challenge : Give an elementary proof that gcd(3 n − 1 , 2 n − 1) 1 /n − → 1 .
Classical Divisibility Sequences 8 Growth Rates of ADS: Another Conjecture Conjecture 1 says that an ADS does not grow too fast. The next conjecture says that for many n , it doesn’t grow at all!
Classical Divisibility Sequences 8 Growth Rates of ADS: Another Conjecture Conjecture 1 says that an ADS does not grow too fast. The next conjecture says that for many n , it doesn’t grow at all! Let A/K be an abelian variety of Conjecture 2. dimension ≥ 2, and let P ∈ A ( K ) be a point such that Z P is Zariski dense in A . Then there is a constant C = C ( A/K, P ) with the property that � ≤ C for infinitely many n ≥ 1. � � ( ∗ ) � N K/ Q D n ( P )
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